Hermann Gross

A maximum likelihood approach to the inverse problem of scatterometry

Scatterometry is frequently used as a non-imaging indirect optical method to reconstruct the critical dimensions (CD) of periodic nanostructures. A particular promising direction is EUV scatterometry with wavelengths in the range of 13 - 14 nm. The conventional approach to determine CDs is the minimization of a least squares function (LSQ). In this paper, we introduce an alternative method based on the maximum likelihood estimation (MLE) that determines the statistical error model parameters directly from measurement data. By using simulation data, we show that the MLE method is able to correct the systematic errors present in LSQ results and improves the accuracy of scatterometry. In a second step, the MLE approach is applied to measurement data from both extreme ultraviolet (EUV) and deep ultraviolet (DUV) scatterometry. Using MLE removes the systematic disagreement of EUV with other methods such as scanning electron microscopy and gives consistent results for DUV.

Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates

Scatterometry as a non-imaging indirect optical method in wafer metrology is also relevant to lithography masks designed for extreme ultraviolet lithography, where light with wavelengths in the range of 13 nm is applied. The solution of the inverse problem, i.e. the determination of periodic surface structures regarding critical dimensions (CD) and other profile properties from light diffraction patterns, is incomplete without knowledge of the uncertainties associated with the reconstructed parameters. The numerical simulation of the diffraction process for periodic 2D structures can be realized by the finite element solution of the two-dimensional Helmholtz equation. The inverse problem can be formulated as a nonlinear operator equation in Euclidean space. The operator maps the sought mask parameters to the efficiencies of diffracted plane wave modes. We employ a Gaus–Newton type iterative method to solve this operator equation and end up minimizing the deviation of the measured efficiency or phase shift values from the calculated ones. We apply our reconstruction algorithm for the measurement of a typical EUV mask composed of TaN absorber lines of about 80 nm height, a period in the range of 420 nm–840 nm, and with an underlying MoSi-multilayer stack of 300 nm thickness. Clearly, the uncertainties of the reconstructed geometric parameters essentially depend on the uncertainties of the input data and can be estimated by various methods. We apply a Monte Carlo procedure and an approximative covariance method to evaluate the reconstruction algorithm. Finally, we analyze the influence of uncertainties in the widths of the multilayer stack by the Monte Carlo method.

Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry

We investigate the impact of line-edge and line-width roughness (LER, LWR) on the measured diffraction intensities in angular resolved extreme ultraviolet (EUV) scatterometry for a periodic line-space structure designed for EUV lithography. LER and LWR with typical amplitudes of a few nanometers were previously neglected in the course of the profile reconstruction. The two-dimensional (2D) rigorous numerical simulations of the diffraction process for periodic structures are carried out with the finite element method providing a numerical solution of the 2D Helmholtz equation. To model roughness, multiple calculations are performed for domains with large periods, containing many pairs of line and space with stochastically chosen line and space widths. A systematic decrease of the mean efficiencies for higher diffraction orders along with increasing variances is observed and established for different degrees of roughness. In particular, we obtain simple analytical expressions for the bias in the mean efficiencies and the additional uncertainty contribution stemming from the presence of LER and/or LWR. As a consequence this bias can easily be included into the reconstruction model to provide accurate values for the evaluated profile parameters. We resolve the sensitivity of the reconstruction from this bias by using simulated data with LER/LWR perturbed efficiencies for multiple reconstructions. If the scattering efficiencies are bias-corrected, significant improvements are found in the reconstructed bottom and top widths toward the nominal values.

First steps towards a scatterometry reference standard

Supported by the European Commission and EURAMET, a consortium of 10 participants from national metrology institutes, universities and companies has recently started a joint research project with the aim of overcoming current challenges in optical scatterometry for traceable linewidth metrology and to establish scatterometry as a traceable and absolute metrological method for dimensional measurements. This requires a thorough investigation of the influence of all significant sample, tool and data analysis parameters, which affect the scatterometric measurement results. For this purpose and to improve the tool matching between scatterometers, CD-SEMs and CD-AFMs, experimental and modelling methods will be enhanced. The different scatterometry methods will be compared with each other and with specially adapted atomic force microscopy (AFM) and scanning electron microscopy (SEM) measurement systems. Additionally novel methods for sophisticated data analysis will be developed and investigated to reach significant reductions of the measurement uncertainties in critical dimension (CD) metrology. To transfer traceability to industrial applications of scatterometry an important step and one final goal of this project is the realisation of different waferbased reference standard materials for calibration of scatterometers. The approaches to reach these goals and first design considerations and preliminary specification of the scatterometry standards are presented and discussed.

Improved grating reconstruction by determination of line roughness in extreme ultraviolet scatterometry

The accurate determination of critical dimensions and roughness is necessary to ensure the quality of photoresist masks that are crucial for the operational reliability of electronic components. Scatterometry provides a fast indirect optical nondestructive method for the determination of profile parameters that are obtained from scattered light intensities using inverse methods. We illustrate the effect of line roughness on the reconstruction of grating parameters employing a maximum likelihood scheme. Neglecting line roughness introduces a strong bias in the parameter estimations. Therefore, such roughness has to be included in the mathematical model of the measurement in order to obtain accurate reconstruction results. In addition, the method allows to determine line roughness from scatterometry. The approach is demonstrated for simulated scattering intensities as well as for experimental data of extreme ultraviolet light scatterometry measurements. The results obtained from the experimental data are in agreement with independent atomic force microscopy measurements.

A scatterometry inverse problem in optical mask metrology

We discuss the solution of the inverse problem in scatterometry i.e. the determination of periodic surface structures from light diffraction patterns. With decreasing details of lithography masks, increasing demands on metrology techniques arise. By scatterometry as a non-imaging indirect optical method critical dimensions (CD) like side-wall angles, heights, top and bottom widths are determined. The numerical simulation of diffraction is based on the finite element solution of the Helmholtz equation. The inverse problem seeks to reconstruct the grating geometry from measured diffraction patterns. The inverse operator maps efficiencies of diffracted plane wave modes to the grating parameters. We employ a Newton type iterative method to solve the resulting minimum problem. The reconstruction quality surely depends on the angles of incidence, on the wave lengths and/or the number of propagating scattered wave modes and will be discussed by numerical examples.

Improved reconstruction of critical dimensions in extreme ultraviolet scatterometry by modeling systematic errors

Scatterometry is a non-imaging indirect optical method that is frequently used to reconstruct the critical dimensions (CD) of periodic nanostructures, e.g. structured wafer surfaces in semiconductor chip production. To solve the inverse problem, we apply a maximum likelihood estimation, introduced in Henn et al (2012 Opt. Express 20 12771–86). Along with the CD values, further relevant quantities like noise parameters of the measured diffraction intensities and the strength of line roughness can be estimated from the measured scattering efficiencies. We investigate three different models for extreme ultraviolet (EUV) scatterometry at an EUV photo mask with increasing complexity by successively including two major sources of systematic errors, namely line roughness and deviations in the multilayer substrate of the EUV mask. Applying the different models to reconstruct the CDs from both simulation and measurement data, we demonstrate the improvements of the reconstruction in terms of simulated and real measurement data. The inclusion of systematic errors in the maximum likelihood approach to the inverse problem leads to a significant reduction of the variances in the estimated CDs implying reduced measurement uncertainty for scatterometry.

EUV and DUV scatterometry for CD and edge profile metrology on EUV masks

To test the applicability of scatterometry on EUV masks we measured a prototype EUV mask both with an EUV scatterometer and a conventional scatterometer operated at 193 nm and compared the results with AFM and CD-SEM measurements provided to us by the mask supplier. The results of both CD-SEM and EUV- and DUV scatterometry show a quite good agreement in linearity despite constant CD offsets for these different metrology tools. The influences of the multilayer and Si capping layer on top of the multilayer thickness on EUV scatterometry results have been modelled with the help of FEM based simulations. A strong correlation has been found between the thickness of the capping layer and the sidewall angle. In general these results demonstrate the applicability both of EUV and DUV scatterometry for the characterisation of absorber structures on EUV masks. The application of DUV scatterometry allows to omit any influence from multilayer features and is only sensitive to the absorber structure. In this way EUV and DUV scatterometry complement each other for metrology on EUV masks. For applications in process optimisation and in process control the use of a conventional VIS/DUV-scatterometer may be sufficient in many cases.

Computational methods estimating uncertainties for profile reconstruction in scatterometry

The solution of the inverse problem in scatterometry, i.e. the determination of periodic surface structures from light diffraction patterns, is incomplete without knowledge of the uncertainties associated with the reconstructed surface parameters. With decreasing feature sizes of lithography masks, increasing demands on metrology techniques arise. Scatterometry as a non-imaging indirect optical method is applied to periodic line-space structures in order to determine geometric parameters like side-wall angles, heights, top and bottom widths and to evaluate the quality of the manufacturing process. The numerical simulation of the diffraction process is based on the finite element solution of the Helmholtz equation. The inverse problem seeks to reconstruct the grating geometry from measured diffraction patterns. Restricting the class of gratings and the set of measurements, this inverse problem can be reformulated as a non-linear operator equation in Euclidean spaces. The operator maps the grating parameters to the efficiencies of diffracted plane wave modes. We employ a Gauss-Newton type iterative method to solve this operator equation and end up minimizing the deviation of the measured efficiency or phase shift values from the simulated ones. The reconstruction properties and the convergence of the algorithm, however, is controlled by the local conditioning of the non-linear mapping and the uncertainties of the measured efficiencies or phase shifts. In particular, the uncertainties of the reconstructed geometric parameters essentially depend on the uncertainties of the input data and can be estimated by various methods. We compare the results obtained from a Monte Carlo procedure to the estimations gained from the approximative covariance matrix of the profile parameters close to the optimal solution and apply them to EUV masks illuminated by plane waves with wavelengths in the range of 13 nm.

On numerical reconstructions of lithographic masks in DUV scatterometry

The solution of the inverse problem in scatterometry employing deep ultraviolet light (DUV) is discussed, i.e. we consider the determination of periodic surface structures from light diffraction patterns. With decreasing dimensions of the structures on photo lithography masks and wafers, increasing demands on the required metrology techniques arise. Scatterometry as a non-imaging indirect optical method is applied to periodic line structures in order to determine the sidewall angles, heights, and critical dimensions (CD), i.e., the top and bottom widths. The latter quantities are typically in the range of tens of nanometers. All these angles, heights, and CDs are the fundamental figures in order to evaluate the quality of the manufacturing process. To measure those quantities a DUV scatterometer is used, which typically operates at a wavelength of 193 nm. The diffraction of light by periodic 2D structures can be simulated using the finite element method for the Helmholtz equation. The corresponding inverse problem seeks to reconstruct the grating geometry from measured diffraction patterns. Fixing the class of gratings and the set of measurements, this inverse problem reduces to a finite dimensional nonlinear operator equation. Reformulating the problem as an optimization problem, a vast number of numerical schemes can be applied. Our tool is a sequential quadratic programing (SQP) variant of the Gauss-Newton iteration. In a first step, in which we use a simulated data set, we investigate how accurate the geometrical parameters of an EUV mask can be reconstructed, using light in the DUV range. We then determine the expected uncertainties of geometric parameters by reconstructing from simulated input data perturbed by noise representing the estimated uncertainties of input data. In the last step, we use the measurement data obtained from the new DUV scatterometer at PTB to determine the geometrical parameters of a typical EUV mask with our reconstruction algorithm. The results are compared to the outcome of investigations with two alternative methods namely EUV scatterometry and SEM measurements.